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A sequence of functions is uniformly Cauchy when, for every $\epsilon >0$, all sufficiently late pairs satisfy

A uniform limit of uniformly continuous maps between metric spaces is uniformly continuous. Approximate the limit at both endpoints by one fixed member of the sequence and use its uniform continuity between them.

Every continuous map from a compact metric space to a metric space is uniformly continuous.

Independently colour the vertices of a countable graph red or blue with equal probabilities. If every vertex has infinite degree, then each fixed vertex has infinitely many neighbours of each colour almost surely. A countable union of the exceptional null events is null, so such a colouring exists.

Take unfriendly partitions of an exhaustion by finite induced subgraphs and use a diagonal subsequence to stabilize the colour of each vertex. Local finiteness makes every neighbourhood stabilize after finitely many steps, so each unfriendly inequality passes to the limit.

Every finite graph has an unfriendly partition. In a maximum cut, moving any vertex to the other side cannot increase the number of crossing edges, which is exactly the required neighbour inequality.

The Stone-Cech compactification $\beta \mathbb{N}$ can be identified with the space of ultrafilters on $\mathbb{N}$. Its basic clopen sets are $\bar{A}=\{\mathcal{U}:A\in \mathcal{U}\}$.

This is the natural-number-space case of Stone-Čech compactification.

A nonprincipal ultrafilter contains no finite set. On an infinite set it contains the cofinite filter.

The principal ultrafilter at $x\in X$ is $\{A\subseteq X:x\in A\}$. An ultraproduct by a principal ultrafilter is isomorphic to the factor indexed by $x$.

For equal masses $m$ and spring constants $k$, vertical displacements from equilibrium have stiffness and mass matricesThe two squared frequencies and displacement ratios areGravity shifts the equilibrium but does not enter these vertical frequencies.

The semiclassical action integralis strictly increasing for a confining one-dimensional potential. Consequently the two-turning-point WKB quantization condition has at most one energy at each quantum number.

Normal ordering places creation operators to the left of annihilation operators. For free fields it removes vacuum self-contractions from a composite operator.

The holomorphic stress-energy tensor generates infinitesimal holomorphic conformal transformations through its operator product expansions.

An operator product expansion expresses the short-distance product of local operators as a sum of local operators multiplied by singular coefficient functions.

A graph-theoretic rhombus is two triangles sharing an edge. Every rhombus-free graph on $n\ge 4$ vertices has at mostedges. If the graph is triangle-free this is Turán's theorem. Otherwise remove a triangle: every remaining vertex has at most one neighbor in it, and induction bounds the edge count by
$\lfloor (n-3{)}^2/4\rfloor +n\le \lfloor n^2/4\rfloor$.

Every $n$-vertex graph containing no $K_{r+1}$ satisfies

If $f\in L^1\cap L^{\infty }$ and $\hat{f}\ge 0$, testing against triangular frequency cutoffs and using monotone convergence gives $\Vert \hat{f}{\Vert }_1\le 2\pi \Vert f{\Vert }_{\infty }$.

A triangle-free graph contains no triangle.

Integer supplies and demands admit an integer optimum because an integer initial basis and every cycle pivot remain integer.

The transportation simplex computes row and column potentials, enters a negative-reduced-cost cell, and pivots around its induced alternating cycle.